| L(s) = 1 | + 3.29·3-s + 2.49·5-s − 1.71·7-s + 7.84·9-s + 0.274·11-s − 1.36·13-s + 8.21·15-s − 0.775·17-s + 3.05·19-s − 5.64·21-s − 4.68·23-s + 1.22·25-s + 15.9·27-s − 2.07·29-s + 6.83·31-s + 0.904·33-s − 4.27·35-s + 8.13·37-s − 4.48·39-s + 5.05·41-s + 9.21·43-s + 19.5·45-s − 10.2·47-s − 4.06·49-s − 2.55·51-s + 3.29·53-s + 0.684·55-s + ⋯ |
| L(s) = 1 | + 1.90·3-s + 1.11·5-s − 0.647·7-s + 2.61·9-s + 0.0827·11-s − 0.378·13-s + 2.12·15-s − 0.188·17-s + 0.700·19-s − 1.23·21-s − 0.977·23-s + 0.244·25-s + 3.06·27-s − 0.385·29-s + 1.22·31-s + 0.157·33-s − 0.722·35-s + 1.33·37-s − 0.718·39-s + 0.789·41-s + 1.40·43-s + 2.91·45-s − 1.49·47-s − 0.580·49-s − 0.357·51-s + 0.452·53-s + 0.0923·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.331858931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.331858931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 113 | \( 1 + T \) |
| good | 3 | \( 1 - 3.29T + 3T^{2} \) |
| 5 | \( 1 - 2.49T + 5T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 - 0.274T + 11T^{2} \) |
| 13 | \( 1 + 1.36T + 13T^{2} \) |
| 17 | \( 1 + 0.775T + 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 23 | \( 1 + 4.68T + 23T^{2} \) |
| 29 | \( 1 + 2.07T + 29T^{2} \) |
| 31 | \( 1 - 6.83T + 31T^{2} \) |
| 37 | \( 1 - 8.13T + 37T^{2} \) |
| 41 | \( 1 - 5.05T + 41T^{2} \) |
| 43 | \( 1 - 9.21T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 3.29T + 53T^{2} \) |
| 59 | \( 1 + 6.09T + 59T^{2} \) |
| 61 | \( 1 - 9.20T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 9.20T + 79T^{2} \) |
| 83 | \( 1 + 0.0523T + 83T^{2} \) |
| 89 | \( 1 - 0.384T + 89T^{2} \) |
| 97 | \( 1 + 1.35T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.931151314481583992976928730774, −7.46498396838077914424975763481, −6.55080522867963260623093824553, −6.03695443832066076117283008586, −4.94585985417181400412772466844, −4.10770424643958765283325522114, −3.39101051247815367862721832344, −2.54945734986030922522443462387, −2.18695465023421442132197275315, −1.12919858059076101812811226443,
1.12919858059076101812811226443, 2.18695465023421442132197275315, 2.54945734986030922522443462387, 3.39101051247815367862721832344, 4.10770424643958765283325522114, 4.94585985417181400412772466844, 6.03695443832066076117283008586, 6.55080522867963260623093824553, 7.46498396838077914424975763481, 7.931151314481583992976928730774
